I've been delving in to baby rudin for the past few weeks and am now trying to finish up chapter 3. However, though I understand the definition he gives for $\lim\sup a_n$, I find it very hard to actually do calculations with it. My non-formal understanding:
If we let A be the set with the limitpoint of all possible subsequences of $\{a_n\}$, then $\lim\sup a_n$ is the supremum of A.
Which is a solid definition, but in order to calculate it - if I'm able to do it at all - I find that I'm having to find the supremum and infinum by looking at the way the series progresses and "eyeballing" a solution. This gets especially problematic when trying to use the ratio and root tests.
So I'm looking for methods to actually start thinking about this in a more practical sense to actually start calculating them.
For reference, this is Rudin's definition:
Given a sequence $\{s_n\}$ of real numbers, let $E$ be the set of numbers $x$ (in the extended real number system) such that $s_{n_k} \to x$ for some subsequence $\{s_{n_k}\}$. This set contains all subsequential limits ..., plus possibly the numbers $+\infty$, $-\infty$. We now ... put $$s^{*} = \sup E,$$ $$s_{*} = \inf E.$$ The numbers $s^{*}$, $s_{*}$ are called the upper and lower limits of $\{s_n\}$; we use the notation $$ \lim_{n \to \infty} \sup s_n = s^{*}, \, \, \, \lim_{n \to \infty} \inf s_n = s_{*}.$$